This paper proposes a preliminary theoretical study for sound ﬁeld and sound environment reproduction in ﬂight vehicles. A fully-coupled cavity, cylindrical shell and exterior radiation model approximates an aircraft cabin mock-up. Material and geometry charateristics are inspired by measurements perfomed on a cabin mock-up. The sound ﬁeld reproduction is based on reproduction error minimization at a microphone array positionned in the cavity. Two reproduction systems, based on actuators or loudspeakers are simulated in order to compare their feasability and performance. The model linking excitator strength with the sound pressure on the spatially extended array region is developped in a matricial form. The promising results obtained in terms of reproduced pressure in the array region in both cases presume the reliability of such dedicated systems.

Sound ﬁeld reproduction applied to ﬂight

vehicles sound environments
C´ edric Camier
1
, Philippe-Aubert Gauthier
1
, Yann Pasco
1
, and Alain Berry
1
1
Universit´ e de Sherbrooke, Sherbrooke, Qu´ ebec, J1K 2R1, Canada
Correspondence should be addressed to C´ edric Camier (Cedric.Camier@USherbrooke.ca)
ABSTRACT
This paper proposes a preliminary theoretical study for sound ﬁeld and sound environment reproduction in
ﬂight vehicles. A fully-coupled cavity, cylindrical shell and exterior radiation model approximates an aircraft
cabin mock-up. Material and geometry charateristics are inspired by measurements perfomed on a cabin
mock-up. The sound ﬁeld reproduction is based on reproduction error minimization at a microphone array
positionned in the cavity. Two reproduction systems, based on actuators or loudspeakers are simulated in
order to compare their feasability and performance. The model linking excitator strength with the sound
pressure on the spatially extended array region is developped in a matricial form. The promising results
obtained in terms of reproduced pressure in the array region in both cases presume the reliability of such
dedicated systems.
1. INTRODUCTION
Since the ﬁrst spatial sound experiments [1], [2], interest
in spatial audio had continuously increased over the past
century [3]. Beside applications to music reproduction
and ﬁlm presentation, spatial sound has recently gained
the attention fromthe transport industry for ﬂight simula-
tors and as a potential sound quality evaluation or design
tool. This paper presents a preliminary theoretical study
for sound ﬁeld and sound environment reproduction in
mock-ups of aircraft cabins.
Since the 1970s, several works have been devoted to the
reproduction or synthesis of exterior and interior noises
of ﬂight vehicles [4]-[7]. Most of these works are primar-
ily devoted to the evaluation of sound quality and annoy-
ance of vehicle noises without any in-depth consideration
of the spatial distribution of sound. However, it is known
that the spatial distribution of sound sources plays an im-
portant role in auditory stream segregation. Indeed, spa-
tial separation of sources reduces masking. Hence, the
spatial distribution of sound should be addressed in cur-
rent work on sound environment reproduction for sound
quality testing or virtual rendering of ﬂight scenarios in
ﬂight vehicle mock-ups and ﬂight simulators. Recent re-
search works go in that direction [8], [9].
1.1. Spatial audio and sound environment re-
production
Most of the recent research works on spatial audio using
multichannel systems are based on few dominant tech-
nologies: stereophonic sound fundamentals extended to
”Surround sound” systems [2], Ambisonics [10] and
wave ﬁeld synthesis (WFS) [11]. Each of which re-
lies on different perceptual and technological hypothe-
sis. Among these technologies, Ambisonics and WFS
are perhaps the twos that have the greatest potential for
psychophysically valid sound environment reproduction.
Ambisonics have already been use for soundscape re-
production [12]. Since these types of sound ﬁeld repro-
duction systems are normally used in more or less well
controlled listening rooms, it has been argued that room
response may degrade the sound ﬁeld reproduction sys-
tem ability to physically recreate and approach the tar-
get sound ﬁeld [13], [14]. Several researchers have then
addressed the spatial room compensation problem [15]-
[22]. The great challenge behind these room compen-
sation methods stands in the requirement that the real
acoustic of the listening room must be replaced by a vir-
tual or target acoustic which have been computed or mea-
sured in an acoustic space different from the listening
room.
1.2. Sound rendering of interior vehicle noise
in vehicle mock-ups
In contrast with generic applications mentioned above,
sound environment or sound ﬁeld reproduction in ve-
hicle mock-ups brings different challenges and some-
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Camier et al. Sound ﬁeld reproduction in aircraft
how simpliﬁes the room compensation issues. Firstly,
the highest quality vehicle sound environment reproduc-
tion system would not only involve a sound system, but
a vehicle mock-up which is visually, mechanically and
geometrically very similar to the real vehicle. More-
over, reproduction sources (either acoustical or vibra-
tional) should be invisible to the listener. Accordingly,
it is expected that the original vehicle and corresponding
mock-up should have a similar, or at least a similar type
of, vibroacoustical behavior. Therefore, room compen-
sation could be more easily applied to that practical case
since the difference between the two systems are greatly
diminished. This have the potential to diminish residual
artifact. Secondly, since many transport applications of
spatial sound are concerned by the physically valid re-
construction of sound ﬁeld, a closed-loop room compen-
sation is mandatory to ensure and physically certify that
the reproduced sound ﬁeld is a physical reconstruction
of the original sound ﬁeld. Indeed, such spatial sound
systems could not rely on the illusory creation of an au-
ditory scene such as achieved in the audio industry since
it might have to be certiﬁed by various agencies, such as
for ﬂight simulators or aircraft sales. Thirdly, most of the
major interior noises are stationary or nearly stationary
(turbulent boundary layer, engine, jet, etc. [23]) so that
any room compensation residual artifact such as pre- and
post-echoes should be inaudible. These three prelimi-
nary hypothesis motivate the interest of room compensa-
tion for ﬂight vehicles sound environment reproduction.
The purpose of this paper is to evaluate the feasibility
of sound ﬁeld reproduction based on room compensa-
tion using a simpliﬁed theoretical model of an aircraft
cabin and mock-up. Several scenarios are compared on
the basis of different reproduction source types: vibra-
tional sources on the cabin structure or acoustical sources
located in the cabin cavity.
1.3. Paper outline
Section 1 introduces the general problem and formu-
lates the main research question addressed in this paper.
Section 2 describes the fully-coupled cavity, cylindri-
cal shell and exterior radiation model that approximates
an aircraft cabin and mock-up. The spatially-extended
sound ﬁeld reproduction method based on reproduction
error minimization is presented in Sec. 3. Simulations
and numerical results are presented and discussed in
Sec. 4. Section 6 gathers the main concluding remarks
and presents future research avenues.
2. CAVITY, SHELL AND EXTERNAL SOUND
FIELD COUPLED MODEL
The vibroacoustic model developped in the following
will be used both for simulating the image pressure ﬁeld
and establishing the inverse model used in reproduction.
It considers a bafﬂed closed cylindrical shell radiating to
interior and exterior spaces. The coupling between the
movement of the shell and the resulting external radia-
tion is taken into account as well as the internal coupling
with the closed acoustic cavity. For the sake of concise-
ness, key points of the model are presented in the sequel.
Full expressions of calculous will be detailed in a future
paper.
2.1. Geometry of the system
As shown in Fig. 1, the 3-component vector displace-
ment u of the cylindrical shell at a given point Q of co-
ordinates (r =a, θ, z) is described by its longitudinal, cir-
cumferential and radial displacements, u, v and w respec-
tively, along the surface S of the shell. At an other point
B which could be situated in the internal volume V
i
or in
the external volume V
e
, both ﬁlled by air of density ρ
0
and characterized by the sound phase speed c, the acous-
tic pressure is noted p.
2.2. Vibroacoustic model
The dynamic of the thin cylindrical shell closed by shear
diaphragms at both ends is governed by the Donnell-
Mushtari theory, referred in [24]. The shell displace-
ment vector discretized onto its in vacuo natural modes
Φ
n
writes:
u(θ, z) =
∞
∑
n=1
A
n
Φ
n
(θ, z) (1)
where A
n
is the n
th
shell modal amplitude and n the set
of modal indexes which refers to nodes in the radial
and circumferential directions and to symmetry type.
Each shell mode shape is normalized with respect to the
modal mass m
n
and is associated to non-dimensional
natural pulsation ω
n
which is given by Leissa [24].
The vibrating shell, immersed into an air-ﬁlled open
space, creates an acoustical radiation which interacts
with its own movement. This coupling could be de-
scribed as an intermodal coupling impedance between
the shell modes [25]. Thanks to an assumption of
inﬁnite-long cylinder, the radiation could be expressed
analytically and then the projection on the ﬁnite-long
surface leads to analytical coupling coefﬁcients of the
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Camier et al. Sound ﬁeld reproduction in aircraft
impedance. This useful approximation has been dis-
cussed and justiﬁed for similar conﬁgurations [26]-[28].
Expression of this impedance is not given here but the
coupling would be represented in matrix form in the fol-
lowing.
The shell internal volume V
i
, also air-ﬁlled, is the seat
of an internal coupling between the shell and the closed
acoustical cavity. Besides, it is our region of interest
since the interior sound ﬁeld would be the target of repro-
duction. Inside the internal volume, the complex sound
pressure ﬁeld p(r, θ, z) is expressed as a linear combina-
tion of real rigid-wall cavity modes Ψ
m
(r, θ, z) [27].
p(r, θ, z) =
∞
∑
m=1
P
m
Ψ
m
(r, θ, z) (2)
where P
m
is the m
th
complex cavity modal amplitude and
m a set of modal indexes which refers to the 3 direc-
tions of space and to the symmetry type. The cavity
mode shapes are normalized with respect to the modal
volume V
m
and associated to ω
m
which are analytically
expressed in [27]. The expressions of coupling coefﬁ-
cients between shell modes and cavity modes are also
analytically known [27]; nevertheless, as the previous
mentionned coupling, one will expressed them in matrix
form only.
Thus, the complete vibroacoustic model written for har-
monic excitations in terms of modal co-ordinates is:
[
C
(cav)
D
(cav,sh)
D
(sh,cav)
C
(sh)
]

[
H
z
]
No
Fig. 2: Natural frequency [Hz] (), sorted in ascending
order, of the rigid-wall cavity and of the coupled system
(•). Schroeder frequency [29] of the rigid-wall cavity is
represented by a horizontal dash-dot line and excitation
frequencies of the two simulations presented (98 Hz and
300 Hz) are plotted in plain lines.
tion systems of a stationary external noise source includ-
ing structural actuators or acoustic excitators close to the
trim panel (here modeled by the shell). Typical simula-
tions will thus involve an exterior plane wave excitating
the dynamic model described in Sec. 2 and virtual mea-
surements of the interior sound ﬁeld by a microphone ar-
ray located in the listening plane of the simpliﬁed mock-
up. With the help of the control method presented in
Sec. 3 performed on the virtual measurements, complex
amplitudes of excitators are deduced to reproduce the tar-
get sound ﬁeld. Then, the error between image sound
ﬁeld (virtually deﬁned in this paper) and reproduction er-
ror is evaluated.
One considers a unitary plane wave of pulsation ω im-
pinging perpendicularly on the shell. The total sound
pressure results in the sum of the incident and the scat-
tered wave ﬁeld [32]. Similarly to the case of ponctual
structural forces in Sec. 2, the total pressure on the shell
surface is projected onto the shell modes to be injected
as F
(sh)
in Eq. (3), then in Eq. (2), in order to obtain the
target image pressure p
(im)
at the microphone array.
Mechanical characteristics and geometrical dimensions
are inspired from measurements performed in a real
mock-up. Particularly, structural damping is computed
from the measured reverberation times. Conﬁguration
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Camier et al. Sound ﬁeld reproduction in aircraft
of the microphone array, actuator positions and speaker
positions have been choosen with respect to fabrication
considerations and in anticipation of the experimental
set-up constraints. Futhermore, one opts for the num-
ber of microphones to be equal to the number of repro-
duction sources in order to have a determined system.
Thus, the microphone array is a a/6 side-length square
composed of 8×8 regularly distributed microphones, the
64-actuator system will denote 2 rows of 2×32 equally-
spaced structural excitators placed on the intersection of
the cylinder with the mid-height plane or with the lis-
tening plane. The listening plane 64-speaker system will
denote 34 equally-spaced radiating monopoles on the lat-
eral edges of the plane combined with 30 equally spaced
monopoles at the ends, see Figs 4, 5, 7 and 8.
Typical Z
(ma)
response is computed for one geometrical
and mechanical conﬁguration of the mock-up. Following
results in terms of inside pressure ﬁeld correspond to two
simulations computed for the same excitating plane wave
except from the selected pulsation. Three factors have
governed our choice of excitation frequency: truncations
of modal bases (to avoid prohibited computation cost),
eigen-frequencies of the coupled system (which guide
the response of the system in low frequency range) and
Schroeder frequency (which is an estimation of the tran-
sition from modal behavior to a diffuse behavior (more
than 3 excited modes for a single frequency)). As shown
in Fig. 2, the ﬁrst excitation frequency is chosen to corre-
spond to one of the ﬁrst eigen-value of the coupled sys-
tem, below the Schroeder frequency. The second one is
situated just above the Schroeder frequency. In fact, to
choose a higher frequency imposes a higher truncation
order in modal bases to insure the convergence of the
solution and so a higher computational cost [31]. The
compromise is arbitrary made to be around 100 modes
for N as well as for M.
Fig. 3 and Fig. 6 show the image sound ﬁeld produced
by an exterior harmonic scattering plane wave of fre-
quency f = 98 Hz and f = 300 Hz, respectively, im-
pinging on the shell in the x
2
axis direction. The printed
sound ﬁeld is thus the complex interior acoustic response
of the vibro-acoustic system which consists in the trun-
cated summation of real modal shapes weighted by the
complex cavity modal amplitudes. As the whole sys-
tem is linear and the excitation is unitary, the visualized
sound ﬁeld could be directly scaled with any excitation
amplitude.
For each two cases of excitation, Figs. 4, 5, 7 and 8
present the results of the sound ﬁeld reproduced by the
two excitator conﬁgurations. In order to evaluate the
quality of reproduction at microphone array location, rel-
ative quadratic errors deﬁned by
e
(m)
q
=
e
(m)
H
e
(m)
p
(m)
H
p
(m)
, (11)
where p
(m)
= p
(im)
(x
(m)
, ω), is computed. Similarly to
this expression, the quadratic error e
(LP)
q
computed on the
whole listening plane will be given.
5. DISCUSSION OF THE RESULTS
The ﬁrst general remark which has to be made is on the
dimension of the image pressure ﬁelds in the cavity. Due
to the unitary exterior acoustical excitation, the sound
amplitude inside the system is very low. It shows the
large-scale relation between exterior and interior sound
for a model dimensionned on measurements on a real
mock-up. Nevertheless, as the complete system is linear,
the physical phenomena of reproduction are well repre-
sented. Secondly, the choice of parameter of regulariza-
tion λ is not motivated here, this study being not the pur-
pose of this paper. One has just to note that for each case,
λ has been adjusted to reproduce correctly both the pres-
sure at the microphone array and the pressure ﬁeld in the
listening plane. A future publication will provide a study
dedicated to this parameter.
For this particular study, one observes that the inverse
method is capable of global reproduction by minimiza-
tion of the error on a discrete local area. In spite of small
differences in the pressure ﬁeld shapes, the two systems
of reproduction provide similar performances in terms of
relative quadratic errors. Nonetheless, the speaker con-
ﬁguration produce generally rougher ﬁeld shape (with
higher slopes) than the actuator conﬁguration. Contrary
to structural excitations of whom the amplitudes result
in the projection on smoother mode shapes in this range
of frequencies, the acoustical excitations involve more
rough acoustic mode shapes because of the strong cou-
pling with rigid-wall cavity modes which are numerous
in the considered frequency range. Because of the mode-
coupling involved in the inverse method, the global shape
of the contributions of the acoustical excitators and con-
sequently the global shape of the reproduced pressure
ﬁeld will show more spatial variations compared to the
actuator conﬁguration for a given truncation.
Preliminary, these ﬁrst feasability results including the
two speciﬁc reproduction systems dedicated to simpliﬁed
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Camier et al. Sound ﬁeld reproduction in aircraft
Fig. 3: Image sound ﬁeld P
(im)
created inside the sys-
tem with a unitary scattering harmonic plane wave of
frequency 98 Hz. Grey surface plotted according to the
x
1
axis in the cylinder represents the scaled pressure in
the listening plane. Pression at the microphone array ()
is also scaled. The pressure scaling factor is equal to
2.09∗10
−13
Pa.
Fig. 4: Sound ﬁeld P
(rep)
reproduced by the 64-actuator
system for an exterior harmonic plane wave of frequency
98 Hz. An acoustical response of actuators on the shell
is given by Eqs. (5) and (6). Origins of stems indicate
the 2-row locations of actuators. Their positive (•) or
negative (◦) amplitudes q
(rep)
oriented towards x
1
for the
top-row and towards −x
1
for the bottom-row are scaled
by 0.7035 N to ﬁt in the graph. The scaling factor is
the same as the respective image ﬁeld. Quadratic error
computations give e
(m)
q
= 0.067 and e
(LP)
q
= 0.41.
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Camier et al. Sound ﬁeld reproduction in aircraft
Fig. 5: Sound ﬁeld P
(rep)
reproduced by the 64-speaker
system for an exterior harmonic plane wave of frequency
98 Hz. Radiation of speakers is modelized by acoustical
reproduction sources (see Eqs. (4) and (6)). Origins of
stems indicate locations of speakers. Their amplitudes
(⋄) q
(rep)
are scaled by 5.3179∗10
−16
m.s
−1
to ﬁt in the
graph. The pressure scaling factor is the same as the re-
spective image ﬁeld plot. Quadratic error computations
give e
(m)
q
= 0.044 and e
(LP)
q
= 0.54.
Fig. 6: Image sound ﬁeld P
(im)
created inside the sys-
tem with a unitary scattering harmonic plane wave of fre-
quency 300 Hz. Grey surface plotted according to the x
1
axis in the cylinder represents the scaled pressure in the
listening plane. Pression at the microphone array () is
also scaled. The sound pressure scaling factor is equal to
1.906∗10
−14
Pa.
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Camier et al. Sound ﬁeld reproduction in aircraft
Fig. 7: Sound ﬁeld P
(rep)
reproduced by a 64-actuator
system for an exterior harmonic plane wave of frequency
300 Hz. Acoustical response of actuators on the shell
is given by Eqs. (5) and (6). Origins of stems indicate
the 2-row locations of actuators. Their positive (•) or
negative (◦) amplitudes q
(rep)
oriented towards x
1
for the
top-row and towards −x
1
for the bottom-row are scaled
by 0.338 N to ﬁt in the graph. The pressure scaling factor
is the same as the respective image ﬁeld plot. Quadratic
error computations give e
(m)
q
= 0.030 and e
(LP)
q
= 0.41.
Fig. 8: Sound ﬁeld P
(rep)
reproduced by a 64-speaker
system for an exterior harmonic plane wave of frequency
300 Hz. Radiation of speakers is modelized by acoustical
monopoles (see Eqs. (4) and (6)). Origins of stems indi-
cate locations of speakers. Their amplitudes (⋄) q
(rep)
are
scaled by 3.1918 ∗ 10
−17
m.s
−1
to ﬁt in the graph. The
sound pressure scaling factor is the same as the respec-
tive image ﬁeld plot. Quadratic error computations give
e
(m)
q
= 0.021 and e
(LP)
q
= 2.75.
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Camier et al. Sound ﬁeld reproduction in aircraft
aircraft space are encouraging. Without any optimization
work, they provide good local reproduction results for
the two exampliﬁed excitation frequencies: 1) below the
Schroeder frequency where the cavity response is dom-
inated by a modal behavior and 2) above the Schroeder
frequency.
6. CONCLUSION
Considering the recent gain of interest of transport in-
dustry for spatial sound, the present paper has proposed
a theoretical formulation of a dedicated sound ﬁeld re-
produced system applied to a simpliﬁed model of ﬂight
aircraft space dimensionned on basis of measurements
in a real mock-up. A complete vibroacoustic model in-
cluding external and internal coupling expanded on the
shell and the rigid-wall cavity modes is used in the in-
verse problem. The spatially-extended sound ﬁeld re-
production method using Tikhonov regularization mini-
mizes the λ-dependent cost function. Two speciﬁc repro-
duction systems have been simulated in order to evaluate
their efﬁciency. The ﬁrst represented lateral trim-panel
actuator system and the second represented enclosing
loudspeaker system. Both of which show good perfor-
mance for the local reproduction and are capable of quite
good global reproduction by ajusting the regularization
parameter.
Recasting these results within the framework of the
project, they provide good expectations for the pratical
method we will use for the reproduction of external-noise
induced sound ﬁeld in the cabin. Indeed, the Z
(ma)
ma-
trix characterizing the vibroacoustic model will be later
computed from measurements in the real system submit-
ted to reproduction excitation and obtained with a mi-
crophone array (presently under construction at GAUS
laboratory). The microphone array would then be re-
moved and the image pressure ﬁeld would be reproduced
by the method described in Sec. 3 for a chosen reproduc-
tion source set-up . The approach using inverse method
is in our case justiﬁed by the type of source signals we
want to reproduce. In fact, for nearly stationary sounds
such as most of the ﬂying aircraft noises, room compen-
sation and equalization residual artifact such as pre- or
post-echoes should be inaudible.
This preliminary study raises some expectations which
will be questionned in a near future. As a ﬁrst prospec-
tive, effect of λ on reproduction error at the microphone
array and in the listening plane should be the object of a
parametric study. More generally, an optimization of the
excitator positions, a more reﬁned vibroacoustic model
computed on a broadband ﬁtting responses of real cabin
mock-up should be the main lines of future work.
7. ACKNOWLEDGMENT
The authors would like to aknowledge
´
Eric Chambatte
for the measurement of mock-up reverberation times.
This work is part of a project involving: Consortium for
Research and Innovation in Aerospace in Qu´ ebec, Bom-
bardier A´ eronautique, CAE, Universit´ e de Sherbrooke
and McGill University, supported by a Natural Sciences
and Engineering Research Council of Canada grant.
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AES 40
TH
INTERNATIONAL CONFERENCE, Tokyo, Japan, October 8–10
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